Optimal packings of bounded degree trees

@article{Joos2019OptimalPO,
  title={Optimal packings of bounded degree trees},
  author={Felix Joos and Jaehoon Kim and Daniel Kuhn and Deryk Osthus},
  journal={Journal of the European Mathematical Society},
  year={2019}
}
We prove that if $T_1,\dots, T_n$ is a sequence of bounded degree trees so that $T_i$ has $i$ vertices, then $K_n$ has a decomposition into $T_1,\dots, T_n$. This shows that the tree packing conjecture of Gyarfas and Lehel from 1976 holds for all bounded degree trees (in fact, we can allow the first $o(n)$ trees to have arbitrary degrees). Similarly, we show that Ringel's conjecture from 1963 holds for all bounded degree trees. We deduce these results from a more general theorem, which yields… 

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References

SHOWING 1-10 OF 44 REFERENCES

Decomposing almost complete graphs by random trees

  • A. Lladó
  • Mathematics, Computer Science
    Electron. Notes Discret. Math.
  • 2014

Packing large trees of consecutive orders

Packing minor-closed families of graphs into complete graphs

Optimal path and cycle decompositions of dense quasirandom graphs

An approximate version of the tree packing conjecture

We prove that for any pair of constants ɛ > 0 and Δ and for n sufficiently large, every family of trees of orders at most n, maximum degrees at most Δ, and with at most (n2) edges in total packs into

Packing Tree Factors in Random and Pseudo-random Graphs

TLDR
It is proved that for a xed tree T on t vertices and > 0, the random graph Gn;p, with high probability contains a family of edge-disjoint T -factors covering all but an -fraction of its edges, as long as 4 np log 2 n.

On the Tree Packing Conjecture

TLDR
The Gyarfas tree packing conjecture is proved and it is proved that any set of trees such that no tree is a star and T_i has n-i+1 vertices packs into K_n (for n) is large enough.

A Fast Approximation Algorithm for Computing the Frequencies of Subgraphs in a Given Graph

In this paper we give an algorithm which, given a labeled graph on $n$ vertices and a list of all labeled graphs on $k$ vertices, provides for each graph $H$ of this list an approximation to the

Packing Trees into the Complete Graph

  • Edward Dobson
  • Mathematics
    Combinatorics, Probability and Computing
  • 2002
TLDR
If T is a tree of order n+1−c′n, c′ [les ] 1/25 (37−8 √21 ) ≈ 0.0135748, such that there exists a vertex x ∈ V(T) and T−x has at least n(1−2c′) isolated vertices, then 2n+1 copies of T may be packed into K2n-1.